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Tourish 的一个根式不等式

陈洪葛 posted @ 12 年前 in 不等式 , 1068 阅读

好久没更新博客了,这段时间比较忙。以后争取有空多贴点好的问题,多谢大家光临我的博客。

 

Problem

If a,b,c are positive real numbers such that a+b+c=3,then
a3+3b+b3+3c+c3+3a6


Proof. By the Cauchy-Schwarz inequality,we have
(a3+3b)(a+3b)(a2+3b)2
Thus,it suffices to show that
a2+3ba+3b6
By Holder inequality,we have
(a2+3ba+3b)2[(a2+3b)(a+3b)][(a2+3b)]3=(a2+9)3
Therefore,it is enough to show that
(a2+9)336(a2+3b)(a+3b)
Let p=a+b+c=3 and q=ab+bc+ca,q3.We have
a2+9=p22q+9=2(9q)
(a2+3b)(a+3b)=a3+3a2b+9a2+3ab=(p33pq+3abc)+3a2b+9(p22q)+3q=10824q+3(abc+a2b)
Since
a2b+b2c+c2a+abc427(a+b+c)3
We get
(a2+3b)(a+3b)24(5q)
Thus.it suffices to show that
(9a)3108(5a)
Which is equivalent to the obvious inequality
(3q)2(21q)0
The equality holds for a=b=c=1.     


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